Continuous-time generalizations as well as filtering theoretic applications of our result will be interesting given the importance of non-degeneracy in facilitating mathematical analysis in the theory of non-linear filtering (see [1]}, [2]}, [3]}, [4]}, [5]} for related, often large-deviations theoretic, studies) in addition to establishing related results for partially observed controlled diffusions (see e.g. [6]} where challenges involving non-degenerate diffusions and relations with viscosity solutions are emphasized). 
